Source: https://jmrplens.github.io/phonometry/vibration/machinery/machine-diagnostics/ # Machine fault frequencies (Norton & Karczub) Condition monitoring begins with arithmetic, not with signal processing. Every rolling-contact bearing, gear pair, induction motor and bladed rotor excites a family of **discrete frequencies fixed by its geometry and its shaft speed**, and knowing where those lines fall is what turns a featureless spectrum into a diagnosis: a peak is only evidence when it sits on a named line. This page covers the kinematics of M. P. Norton and D. G. Karczub, *Fundamentals of Noise and Vibration Analysis for Engineers* (2nd ed., Cambridge University Press, 2003), Section 8.4, and how they combine with the signal chain the library already provides - the [envelope spectrum and cepstrum](https://jmrplens.github.io/phonometry/signals/spectra/cepstrum-echoes/) and [time synchronous averaging](https://jmrplens.github.io/phonometry/signals/spectra/synchronous-averaging/). The working view is the two put together: the envelope spectrum of a band-passed vibration record with the predicted lines drawn on top and named. *The four tall lines land on BPFO and its harmonics; nothing appears on BPFI or BSF. That is the diagnosis: a spall on the outer race.*

Show the code for this figure

```python import matplotlib.pyplot as plt import numpy as np from phonometry import bearing_fault_frequencies, envelope_spectrum, noise_signal # The bearing: 15 rollers on a 34 mm pitch diameter, 6 mm rollers, # 12.96 degrees contact angle, shaft at 2000 r/min. faults = bearing_fault_frequencies(2000.0, 15, 6.0, 34.0, contact_angle_deg=12.96) bpfo, shaft = faults["BPFO"], faults.shaft_rate # 207.0 Hz, 33.33 Hz # A spalled outer race: one impact per BPFO period, each ringing a 3 kHz # housing resonance, load-modulated once per revolution, buried in noise. fs, seconds = 20000.0, 2.0 t = np.arange(int(fs * seconds)) / fs impacts = np.zeros_like(t) for k in range(int(seconds * bpfo)): impacts[round(k / bpfo * fs)] = 1.0 + 0.35 * np.cos( 2.0 * np.pi * shaft * k / bpfo) tau = np.arange(int(0.004 * fs)) / fs ring = np.exp(-tau / 6.0e-4) * np.sin(2.0 * np.pi * 3000.0 * tau) x = np.convolve(impacts, ring)[: t.size] * 0.6 x += 0.35 * np.sin(2.0 * np.pi * shaft * t) # residual unbalance x += noise_signal(fs, seconds, color="white", rms=0.25, seed=17) # Band-pass the resonance the impacts ring, envelope it, transform it, # and let the result draw its own lines on top. spectrum = envelope_spectrum(x, fs, band=(2000.0, 4000.0)) faults.within(1.0, 5.0 * bpfo).plot(spectrum=spectrum) plt.show() ```

## 1. Rolling-contact bearings Norton reproduces the eleven discrete frequencies collected by Shahan and Kamperman (Eqs. 8.4 to 8.14). With the shaft speed $N$ in r/min, $Z$ rolling elements of diameter $d$ on a pitch diameter $D$, a contact angle $\phi$, and writing $g = (d/D)\cos\phi$ and $f_\mathrm{s} = N/60$: | Name | Meaning | Expression | Norton eq. | |---|---|---|---| | `shaft` | shaft rotational frequency | $f_\mathrm{s} = N/60$ | (8.4) | | `FTF` | cage (fundamental train) frequency | $(f_\mathrm{s}/2)(1 - g)$ | (8.5) | | `FTF_rel` | cage relative to the rotating race | $f_\mathrm{s} - \text{FTF}$ | (8.11) | | `BSF` | rolling-element rotational frequency | $(f_\mathrm{s}/2)(D/d)(1 - g^2)$ | (8.7) | | `BDF` | rolling-element spin frequency | $2\,\text{BSF}$ | (8.10) | | `BPFO` | element pass frequency, outer race | $Z(f_\mathrm{s}/2)(1 - g)$ | (8.8) | | `BPFI` | element pass frequency, inner race | $Z(f_\mathrm{s}/2)(1 + g)$ | (8.9) | Three of them carry the diagnosis: `BPFO` for an outer-race defect, `BPFI` for an inner-race defect, and `BDF` for a damaged rolling element or cage. Two identities are worth remembering because they are exact and catch data-entry errors instantly: $\text{BPFO} + \text{BPFI} = Z f_\mathrm{s}$ always, and $\text{BPFO} = Z \times \text{FTF}$ whenever the outer race is stationary. ```python from phonometry import bearing_fault_frequencies res = bearing_fault_frequencies(2000.0, 15, 6.0, 34.0, contact_angle_deg=12.96) print(round(res["BPFO"], 1), round(res["BPFI"], 1)) # 207.0 293.0 print(round(res["BPFO"] + res["BPFI"], 1)) # 500.0 = 15 x 33.33 print(res.as_dict()) ``` Only the ratio $d/D$ enters, so the two diameters may be given in millimetres, inches or metres as long as both use the same unit. `rotating_race="outer"` moves the cage to Eq. (8.6); `BPFO` and `BPFI` do not move, because Norton's Eqs. (8.8) and (8.14), and (8.9) and (8.13), are pairwise identical. These are the kinematics of a **pure rolling contact**. A real bearing under load slips a little, so the measured lines wander by one or two per cent from the prediction; that is normal and is why the overlay is read as "the peak is on the line", not "the peak is at the line". ## 2. Gears The gear-meshing (tooth-passing) frequency of a wheel with $N$ teeth is $\text{GMF} = N f_\mathrm{s}$ (Eq. 8.3), with integer harmonics. What distinguishes the fault types is the **sideband** family around each harmonic: a single chipped tooth or an eccentric wheel modulates the mesh once per revolution and produces low, flat sidebands spaced by the shaft rate, while distributed wear raises tall sideband groups and lifts the higher mesh harmonics. ```python from phonometry import gear_mesh_frequencies # A 28-tooth pinion on a 1500 r/min shaft, with two sideband orders. res = gear_mesh_frequencies(1500.0, 28, harmonics=3, sidebands=2) print(round(res["GMF"], 1)) # 700.0 print(round(res["GMF+1x"], 1)) # 725.0 (GMF + one shaft order) print(res.harmonics("GMF", 3)) # [ 700. 1400. 2100.] ``` Pass `sideband_rate` to modulate at the *mating* wheel's shaft rate instead, which is what a fault on the other wheel of the pair produces. ## 3. Induction motors Three lines are always present in a motor bearing signal: `1x` (mechanical unbalance), `2x` (misalignment with the driven load) and `2fe` (a non-uniform air gap, torque pulses and the winding and rotor-bar electrical faults). Rotor defects that produce static or dynamic air-gap eccentricity are read on the **rotor-slot harmonics** of Eq. (8.20), $$ f_\mathrm{sh} = f_\mathrm{e} \left[(2R/p)(1 - s) \pm 2(n - 1)\right] $$ for $R$ rotor bars, $p$ magnetic poles (not pole pairs), unit slip $s$ and $n = 1, 2, \ldots$. Dynamic eccentricity dresses the dominant slot harmonic with sidebands at $\pm$ the shaft rate and $\pm$ the slip frequency. ```python from phonometry import induction_motor_frequencies # Sixty rotor bars, six magnetic poles, 3600 r/min, no slip. res = induction_motor_frequencies(3600.0, 6, 60, slip=0.0) print(round(res["1x"]), round(res["2x"]), round(res["2fe"]), round(res["fsh"])) # 60 120 360 3600 ``` Give the slip directly, or give the mains `supply_frequency` and let the slip follow from it and the measured shaft speed. The supply frequency is taken as $f_\mathrm{e} = f_\mathrm{s}\,p / (2(1 - s))$; Norton's Eq. (8.19) writes `fe = fs p / 2`, which is the same expression at zero slip but does not reduce Eq. (8.20) to the physical rotor-bar passing rate $R f_\mathrm{s}$ when the machine is loaded. Under load the result also carries the slip frequency and the pole-pass line $\text{FP} = p \times f_\text{slip}$, the sideband spacing that a broken rotor bar puts around `1x`. ## 4. Fans, blowers and pumps The blade-passing frequency of an impeller with $N$ blades is $f_\mathrm{b} = n N f_\mathrm{s}$ (Eq. 8.15); a pump's hydraulic pulsations follow the same form with $N$ pumping events per revolution (Eq. 8.18), and a rotary positive-displacement blower repeats four times per revolution, so it is handled by passing $4 \times \text{r/min}$. In a **ducted axial fan** the blades also interact with the stator vanes and set up rotating pressure patterns with $m_\mathrm{L} = nN \pm kV$ lobes for $V$ vanes (Eq. 8.16), turning at $n N f_\mathrm{s} / m_\mathrm{L}$ (Eq. 8.17). A pattern that spins faster than the blades themselves radiates far more strongly, which is why the choice of $N$ and $V$ matters, and why those speeds are the ones to check against the duct cut-on frequencies. Each line is named `lobe n= m=`: the blade harmonic belongs in the name because Eq. (8.17) carries $n$, so the same lobe count reached from a different harmonic is a different pattern turning at a different speed. ```python from phonometry import blade_pass_frequencies # Six blades, four vanes, 3500 r/min. res = blade_pass_frequencies(3500.0, 6, harmonics=1, n_vanes=4) print(round(res["BPF"])) # 350 print(round(res["lobe n=1 m=2"]), round(res["lobe n=1 m=10"])) # 175 35 ``` ## 5. From a raw record to a named line The kinematics answer *where to look*; the signal chain answers *what is there*. For a bearing the classical route has three steps, and the library has one function for each: 1. **Isolate the resonance the impacts ring.** A spall excites a high-frequency structural resonance of the housing, far above the shaft harmonics and the gear mesh. Pass its band as `band=(low, high)` to `envelope_spectrum`, which band-passes the record with a zero-phase filter before the detector so the modulation phase is untouched. 2. **Detect and transform the envelope.** The repetition rate of the impacts is an amplitude modulation of that resonance, so it becomes a *line* in the envelope spectrum, at the fault frequency and its harmonics. See the [cepstrum and envelope-spectrum guide](https://jmrplens.github.io/phonometry/signals/spectra/cepstrum-echoes/). 3. **Name the lines.** Overlay the kinematic prediction. `.plot(spectrum=…)` accepts any `envelope_spectrum` result (or anything exposing `frequencies` and `amplitude`) and draws the predicted lines over it, coloured by family so shaft harmonics never read as bearing evidence. Two more tools help when the record is crowded. [Time synchronous averaging](https://jmrplens.github.io/phonometry/signals/spectra/synchronous-averaging/) removes everything not locked to one shaft, which separates a gearbox's two shafts before the sidebands are read; and the [cepstrum](https://jmrplens.github.io/phonometry/signals/spectra/cepstrum-echoes/) collapses a whole harmonic or sideband family onto a single quefrency spike, which is the fastest way to tell *which* periodicity dominates when several are superimposed. ```python from phonometry import ( bearing_fault_frequencies, combine_fault_lines, envelope_spectrum, gear_mesh_frequencies, ) record, fs = x, 20000.0 # the housing record and its sample rate shaft_rpm = 2000.0 # from the tacho, not from the nameplate # The bearing's own lines, the mesh family of the pinion it supports, and the # shaft harmonics. Put them all on one axes. bearing = bearing_fault_frequencies(shaft_rpm, 15, 6.0, 34.0, contact_angle_deg=12.96) gear = gear_mesh_frequencies(shaft_rpm, 28, harmonics=2, sidebands=1) lines = combine_fault_lines(bearing, gear) spectrum = envelope_spectrum(record, fs, band=(2000.0, 4000.0)) lines.within(1.0, 1600.0).plot(spectrum=spectrum) ``` `within(low, high)` trims the family to the span the analysis actually covers, which matters because the highest mesh harmonic is usually far above the useful part of an envelope spectrum. ## Covered and not covered **Covered.** The kinematic families of Norton Section 8.4: rolling-contact bearings (Eqs. 8.4 to 8.14), gear mesh and sidebands (Eq. 8.3), induction-motor supply, slip, pole-pass and rotor-slot lines (Eqs. 8.19 and 8.20), and blade-passing frequencies with the lobed interaction patterns of a ducted axial fan (Eqs. 8.15 to 8.18), through `bearing_fault_frequencies`, `gear_mesh_frequencies`, `induction_motor_frequencies`, `blade_pass_frequencies`, `combine_fault_lines` and `shaft_rate`. **Not covered.** These are *predictions*, not detections: nothing here decides whether a line is present, only where it would be. The amplitude criteria that turn a present line into a severity assessment (crest factor and kurtosis trends, ISO 10816 / ISO 20816 velocity severity bands) are outside this module; Norton's own guidance is qualitative (crest factors of 2.5 to 3.5 for a healthy bearing, up to about 11 with impulsive defects; kurtosis near 3 healthy and above 4 damaged). The pole-pass frequency $\text{FP} = p \times f_\text{slip}$ is standard condition-monitoring practice rather than Norton's: he gives the slip frequency itself as the broken-bar sideband spacing and does not multiply it by the pole count. ## See also - [Cepstrum, echoes and the envelope spectrum](https://jmrplens.github.io/phonometry/signals/spectra/cepstrum-echoes/): the envelope spectrum these lines are drawn on, and the cepstral collapse of a harmonic family. - [Time synchronous averaging](https://jmrplens.github.io/phonometry/signals/spectra/synchronous-averaging/): separating one shaft's signature from everything asynchronous with it. - [Mechanical mobility and the FRF family (ISO 7626-1)](https://jmrplens.github.io/phonometry/vibration/structural/mechanical-mobility/): finding the structural resonances a defect impact excites. - [Bending-wave transmission at plate junctions](https://jmrplens.github.io/phonometry/vibration/structural/junction-transmission/): what happens to the vibration once it leaves the machine. ## References - Norton, M. P., & Karczub, D. G. (2003). *Fundamentals of noise and vibration analysis for engineers* (2nd ed., Section 8.4). Cambridge University Press. See the [bibliography](https://jmrplens.github.io/phonometry/reference/bibliography/) for full entries.